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FFT and orthogonal discrete transform on weight lattices of semi-simple Lie groups

Bastian Seifert

TL;DR

The paper develops two algebro-geometric-inspired routes to fast Fourier transforms for polynomial algebras in several variables within algebraic signal processing. It extends induced-module and decomposition-based FFT derivations to multivariate settings and introduces a generalized Gauss–Jacobi orthogonalization via a multivariate Christoffel–Darboux formula, linking to Gaussian cubature under suitable conditions. By connecting multivariate Chebyshev polynomials to weight lattices of semisimple Lie groups, the work yields fast transforms for lattices such as $A_2$ and $C_2$, including a directed hexagonal lattice example, and discusses the role of radical ideals, Gröbner bases, and transversal decompositions in achieving computational efficiency. The results broaden algebraic-signal-processing methods beyond square lattices, enabling orthogonal and non-orthogonal multivariate transforms that exploit Lie-theoretic structures. Overall, the paper provides a unified, geometry-informed framework for deriving fast multivariate transforms with concrete lattice applications.

Abstract

We give two algebro-geometric inspired approaches to fast algorithms for Fourier transforms in algebraic signal processing theory based on polynomial algebras in several variables. One is based on module induction and one is based on a decomposition property of certain polynomials. The Gauss-Jacobi procedure for the derivation of orthogonal transforms is extended to the multivariate setting. This extension relies on a multivariate Christoffel-Darboux formula for orthogonal polynomials in several variables. As a set of application examples a general scheme for the derivation of fast transforms of weight lattices based on multivariate Chebyshev polynomials is derived. A special case of such transforms is considered, where one can apply the Gauss-Jacobi procedure.

FFT and orthogonal discrete transform on weight lattices of semi-simple Lie groups

TL;DR

The paper develops two algebro-geometric-inspired routes to fast Fourier transforms for polynomial algebras in several variables within algebraic signal processing. It extends induced-module and decomposition-based FFT derivations to multivariate settings and introduces a generalized Gauss–Jacobi orthogonalization via a multivariate Christoffel–Darboux formula, linking to Gaussian cubature under suitable conditions. By connecting multivariate Chebyshev polynomials to weight lattices of semisimple Lie groups, the work yields fast transforms for lattices such as and , including a directed hexagonal lattice example, and discusses the role of radical ideals, Gröbner bases, and transversal decompositions in achieving computational efficiency. The results broaden algebraic-signal-processing methods beyond square lattices, enabling orthogonal and non-orthogonal multivariate transforms that exploit Lie-theoretic structures. Overall, the paper provides a unified, geometry-informed framework for deriving fast multivariate transforms with concrete lattice applications.

Abstract

We give two algebro-geometric inspired approaches to fast algorithms for Fourier transforms in algebraic signal processing theory based on polynomial algebras in several variables. One is based on module induction and one is based on a decomposition property of certain polynomials. The Gauss-Jacobi procedure for the derivation of orthogonal transforms is extended to the multivariate setting. This extension relies on a multivariate Christoffel-Darboux formula for orthogonal polynomials in several variables. As a set of application examples a general scheme for the derivation of fast transforms of weight lattices based on multivariate Chebyshev polynomials is derived. A special case of such transforms is considered, where one can apply the Gauss-Jacobi procedure.

Paper Structure

This paper contains 2 sections, 1 theorem, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. An algebro-geometric perspective on signal processing and FFT

Key Result

theorem 1

Let $\mathscr{A}$ be an algebra with subalgebra $B$. Let $M_t = \Pi_{}^{d}(y) / J_t$ be a set of $\mathscr{B}$-modules such that $N = \bigoplus_{t \in T} t \mathbin{\triangleright} M_t = \Pi_{}^{n} / I$, with $T = \{t_1,\dots,t_w\} \subset \mathscr{A}$ a finite set, is an $\mathscr{A}$-module. Assum where $B_{\bigoplus b_t}^{b_N}$ is the basis change from the basis $b_N$ to the concatenation of ba

Figures (2)

  • Figure 1: Visualization of the finite, discrete time signal model.
  • Figure 2: Sections of a vector bundle (red) over points on a circle (blue) form the signals of the finite, discrete time signal model.

Theorems & Definitions (3)

  • remark thmcounterremark
  • theorem 1: FFT algorithms, bottom-up approach
  • proof